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Gamma Distribution Calculator

Use this calculator to compute probabilities, percentiles, and visualize the Gamma distribution. Input the shape and scale parameters, and the specific value(s) to calculate left, right, between, or outside probabilities.

Shape Parameter (k):

Scale Parameter (θ):

Point 1 (x₁):

Point 2 (optional for between/outside):

Below Value Above Value Between Two Values Outside Two Values Show CDF

Calculated Probability:

Understanding the Gamma Distribution

The Gamma distribution is a continuous probability distribution used to model the time until an event occurs a certain number of times. It is commonly applied in fields like survival analysis, queuing models, and insurance risk.

Key Components of the Gamma Distribution

Shape Parameter (k): Defines the shape of the distribution. Higher values of \( k \) result in a more symmetric distribution.
Scale Parameter (θ): This stretches or compresses the distribution horizontally. It is also important to note that the **rate parameter** \( \gamma \) is the inverse of the scale parameter, meaning \( \gamma = 1/\theta \).

Gamma Distribution Formula

The probability density function (PDF) for the Gamma distribution is given by the formula:

\[ f(x; k, \theta) = \frac{x^{k-1} e^{-\frac{x}{\theta}}}{\theta^k \Gamma(k)}, \quad \text{for} \ x > 0 \]

Conditions for Using the Gamma Distribution

The Gamma distribution applies under specific conditions:

Non-Negativity: The Gamma distribution is only defined for non-negative values, meaning that \( X \geq 0 \).
Shape and Scale Parameters: The shape parameter \( k \) and the scale parameter \( \theta \) must be positive real numbers.

Step-by-Step Example: Finding the Probability

Suppose we want to calculate the probability that a random variable \( X \), following a Gamma distribution with shape parameter \( k = 2 \) and scale parameter \( θ = 3 \), takes a value less than \( x = 5 \).

Step 1: Formula Setup

The probability we are looking for is \( P(X < 5) \). This is given by the cumulative distribution function (CDF) of the Gamma distribution:

\[ P(X < 5) = \int_0^5 \frac{x^{k-1} e^{-\frac{x}{\theta}}}{\theta^k \Gamma(k)} \, dx \]

Step 2: Plug in the Values

Substituting \( k = 2 \) and \( \theta = 3 \) into the formula, we get:

\[ P(X < 5) = \int_0^5 \frac{x^{1} e^{-\frac{x}{3}}}{3^2 \Gamma(2)} \, dx \]

Here, \( 3^2 = 9 \) and the Gamma function \( \Gamma(2) \) is evaluated as \( \Gamma(2) = (2 - 1)! = 1! = 1 \). The Gamma function \( \Gamma(n) \) is a generalization of the factorial function, where for positive integers, it is equivalent to \( (n-1)! \).

\[ P(X < 5) = \int_0^5 \frac{x^{1} e^{-\frac{x}{3}}}{9 \times 1} \, dx = \int_0^5 \frac{x e^{-\frac{x}{3}}}{9} \, dx \]

Step 3: Compute the Integral

After evaluating the integral, we find:

\[ P(X < 5) \approx 0.4963 \]

Gamma Distribution Probability Calculator